Badiale M., Serra E. Semilinear Elliptic Equations for Beginners: Existence Results via the Variational Approach

Подождите немного. Документ загружается.

4.1 Deformations 151

Proof We construct a deformation using a variant of (4.5) to have the ﬂow lines

deﬁned for all times. We solve the problem

⎧

⎨

⎩



(t, u) =−

∇J(η(t,u))

1 +∇J(η(t,u))

η(0,u)=u,

(4.6)

where η



denotes differentiation with respect to t.

If we call F(η) the right-hand side of (4.6), we see that F : H → H is locally

Lipschitz, because ∇J is so, and of course F(u)≤1 for all u. We can then apply

Theorem 4.1.14 to obtain a solution η(t, u) of problem (4.6) deﬁned in all of R.

By deﬁnition we have η(0,u)=u for all u, and since

J(η(t,u))=



∇J(η(t,u))





(t, u)



=−

∇J(η(t,u))

1 +∇J(η(t,u))

≤0,

the fact that u is in some J

implies η(t,u) ∈J

for all t ≥0.

We now want to prove that η can be used to construct a deformation of J

in J

For every u ∈J

, we deﬁne

T(u)=sup{t ≥0 |η(t,u) ∈J

This could be inﬁnite for some u, in case the ﬂow line η(·,u) never enters J

Notice that, by continuity, T(u)>0 for every u ∈J

, and for every number

t ∈[0, T (u)),wehaveη(t, u) ∈J

We claim that T(u)is not only ﬁnite for every u, but also uniformly bounded, in

the sense that

sup{T(u)|u ∈J

}< +∞. (4.7)

To see this, we ﬁrst notice that

there exists δ>0 such that ∇J(u)≥δ for all u ∈J

. (4.8)

Indeed, if (4.8) were false, then there would be a sequence {u

}

⊂J

such that

∇J(u

)≤1/k. Passing to a subsequence, we would get J(u

) → c ∈[a,b] and

∇J(u

) →0; so {u

} would be a Palais–Smale sequence at level c ∈[a,b], and this

is impossible by assumption.

To go back to the proof of (4.7), ﬁx u ∈ J

and take t ∈[0, T (u)). Then for

every s ∈[0,t] there results η(s,u) ∈J

, so that

∇J(η(s,u))≥δ for all s ∈[0,t].

Since the function x →−

1+x

is decreasing for x ≥0, this implies

−

∇J(η(s,u))

1 +∇J(η(s,u))

≤−

1 +δ

and this holds for every s ∈[0,t]. Hence

J(η(t,u))−J(η(0,u))≤



J(η(s,u))ds =





−

∇J(η)

1 +∇J(η)



≤





−

1 +δ



ds =−t

1 +δ

152 4 Introduction to Minimax Methods

So, for every t ∈[0, T (u)),

a<J(η(t,u))≤ J(η(0,u))−t

1 +δ

=J(u)−t

1 +δ

≤b −t

1 +δ

namely,

b −a

(1 +δ).

Setting μ =

b−a

(1 + δ), we have proved that for every t ∈[0, T (u)), there results

t<μ, which of course implies T(u)≤μ, and this holds for all u ∈J

. Property

(4.7) is proved.

Notice also that, by continuity, J(η(T(u),u))=a.

We now deﬁne ˜η :[0, 1]×J

→J

˜η(t, u) =η(μt,u).

From the previous discussion ˜η enjoys the following properties:

•˜η(0,u)=η(0,u)= u for all u ∈J

•˜η(t, u) =η(μt,u) ∈J

for all u ∈J

• J(˜η(1,u))=J(η(μ,u))≤J(η(T(u),u))≤a for all u ∈J

As ˜η is continuous, ˜η is a deformation of J

that deforms J

in J

. 

In most of the applications we will need a “local” version of the preceding result.

Local here means that we only look at the functional around a certain level set.

Lemma 4.1.18 Assume that H is a Hilbert space, let J ∈C

1,1

(H ), and let c ∈ R.

Assume that there is no Palais–Smale sequence at level c. Then for every ε>0 small

enough, the set J

c+ε

is deformable in J

c−ε

with a deformation that ﬁxes J

c−2ε

Proof We have to ﬁnd a continuous η :[0, 1]×H →H such that

• η(0,u)=u for all u,

• η(t, u) =u for all u ∈J

c−2ε

and for all t,

• η(t, u) ∈J

c−ε

for all u ∈J

c−ε

and for all t,

• η(1,u)∈J

c−ε

for all u ∈J

c+ε

To prove all this, we start by claiming that

there exist ε

> 0 and δ>0 such that

∇J(u)≥δ for every u ∈J

c+ε

c−ε

Indeed, if this is false then we can ﬁnd positive numbers ε

→ 0, δ

→ 0 and a

sequence u

∈J

c+ε

c−ε

such that ∇J(u)≤δ

. In other words, the sequence

} satisﬁes

J(u

) →c and ∇J(u

) →0inH,

namely it is a Palais-sequence at level c, which is impossible by assumption.

We now pick ε ∈(0,ε

), and take a “cut-off” function ψ ∈C

∞

(R) such that

4.2 The Minimax Principle 153

• 0 ≤ψ(s)≤1 for all s,

• ψ(s)=0in(−∞,c−2ε],

• ψ(s)=1in[c −ε, c +ε].

We then solve the problem

⎧

⎨

⎩



(t, u) =−ψ(J (η(t, u)))

∇J(η(t,u))

1 +∇J(η(t,u))

η(0,u)=u.

(4.9)

Once again, the right-hand side of (4.9), is a locally Lipschitz continuous function

of η and is uniformly bounded on H . Then the ﬂow η is well deﬁned and continuous

in R ×H .

Noticing that ψ(J(u)) = 0inJ

c−2ε

, we have that η(t,u) = u for all u ∈ J

c−2ε

and all t, namely η ﬁxes J

c−2ε

The other properties can be deduced observing that ψ(J(u)) = 1 whenever

c − ε ≤ J(u) ≤ c + ε, so that one can use Lemma 4.1.17 with a and b replaced

by c −ε and c +ε respectively. 

The previous lemma shows that the following alternative holds: either there is

a Palais–Smale sequence at level c, or for every ε>0 small enough, the sublevel

c+ε

is deformable in J

c−ε

(and moreover with a deformation that “ﬁxes” J

c−2ε

Remark 4.1.19 There are many versions of deformations lemmas in the literature,

including some very precise ones, see for example [43, 45]. For our purposes, the

two above lemmas are sufﬁcient, and are simpler than the general results.

Remark 4.1.20 In many textbooks the assumptions of the previous lemma are stated

in the following equivalent form: assume that there are no critical points at level c

and that (PS)

holds. We prefer the statement of Lemma 4.1.18 because it highlights

the central role of Palais–Smale sequences. One can then summarize the principle

by saying that the only obstruction to deformations is the presence of Palais–Smale

sequences.

4.2 The Minimax Principle

We now introduce the minimax procedure, namely the main object of this chapter.

We will do it at a rather abstract level, which has the advantage of producing a very

ﬂexible tool which can be adapted to a variety of situations, even beyond the aims

of this book.

We begin with some deﬁnitions.

Deﬁnition 4.2.1 Let H be a Hilbert space and let η :[0, 1]×H → H be a defor-

mation of H . A family  of subsets of H is said to be invariant for η if

for every A ∈ and every t ∈[0, 1], there results η(t,A) ∈.

154 4 Introduction to Minimax Methods

Deﬁnition 4.2.2 Let H be a Hilbert space and let J : H → R be a functional.

A family  of subsets of H is called a minimax class. The value

c = inf

A∈

sup

u∈A

J(u)

is called the minimax level associated to ; we also say that  works at level c.

Notice that it may happen that c =+∞or c =−∞.

Finally we have

Deﬁnition 4.2.3 Let H be a Hilbert space, J :H → R a functional,  a minimax

class, and c its minimax level. We say that  is admissible with respect to J if

1. c ∈R,

2. for every ε>0 small enough,  is invariant for all deformations that ﬁx J

c−2ε

This last deﬁnition contains a rather subtle point. Indeed the deformations for

which a class  should be invariant depend on c, which depends on .

So the deformations for which a class  should be invariant depend on the class

itself. This seems at ﬁrst rather confusing, but we will see in the applications that

the difﬁculty disappears as soon as the number c is determined by good topological

properties. The second request in Deﬁnition 4.2.3 is extremely useful, because it

reduces the number of deformations for which  must be invariant.

For the moment we go on at an abstract level with the main result of Critical

Point Theory.

Theorem 4.2.4 (Minimax principle) Let H be a Hilbert space and let J ∈C

1,1

(H ).

Let  be an admissible minimax class at level c. Then there exists a Palais–Smale

sequence for J at level c. If J satisﬁes (PS)

, then there exists a critical point for J

at level c.

Proof Assume that there is no Palais–Smale sequence at level c. Then we can

choose ε>0 so small that Lemma 4.1.18 and 2. of Deﬁnition 4.2.3 simultaneously

apply. This means that J

c+ε

is deformable in J

c−ε

with a deformation η that ﬁxes

c−2ε

and that  is invariant for this η.

By deﬁnition of minimax level, there exists a set A ∈ such that

sup

u∈A

J(u)≤c +ε,

that is, A ⊆J

c+ε

. Applying the deformation η to A we obtain

η(1,A)⊂η(1,J

c+ε

) ⊂J

c−ε

which implies

sup

u∈η(1,A)

J(u)≤c −ε.

4.3 Two Classical Theorems 155

On the other hand, as  is invariant for η, we have that η(1,A) ∈ , so that by

deﬁnition of c,

sup

u∈η(1,A)

J(u)≥c.

This contradiction proves that there must be a Palais–Smale sequence at level c.The

second part of the theorem is obvious, see Lemma 4.1.8. 

We will see some classical and particular cases of this principle in the next sec-

tions. We just point out that this principle is quite general and uniﬁes different par-

ticular approaches that appeared separately in the literature.

Even in “trivial” contexts, it yields some nontrivial consequences, as the next

example shows.

Example 4.2.5 Let H be a Hilbert space and let J ∈ C

1,1

(H ). We consider the

“trivial” minimax class

 =



{u}|u ∈H



namely the class of subsets of H consisting of a single point. This class is obviously

invariant for every deformation η, since η(t, {u}) is a singleton for every t, and

hence it still belongs to . Thus the class  is admissible if and only if its minimax

level c is ﬁnite. But

c = inf

A∈

sup

u∈A

J(u)= inf

{u}∈

sup

u∈{u}

J(u)= inf

u∈H

J(u),

so that the class is admissible if and only if J is bounded from below.

The new, often very important information provided by the minimax principle,

is the existence not only of a minimizing sequence, but of a Palais–Smale sequence

at level c = inf

J . Thus, from now on, whenever a C

1,1

functional J is bounded

from below, we can conclude that there exists a minimizing sequence {u

}

with the

further noteworthy property that

∇J(u

) →0inH.

This property has a nontrivial proof even for real functions of one variable.

4.3 Two Classical Theorems

In this section we present the two most celebrated results of Critical Point Theory.

We will see how they both ﬁt into the framework given by the minimax principle.

These two results have been generalized in all possible directions, but are still used

very much in their original form.

Theorem 4.3.1 (Mountain Pass Theorem [5]) Let H be a Hilbert space, and let

J ∈ C

1,1

(H ) satisfy J(0) = 0. Assume that there exist positive numbers ρ and α

such that

156 4 Introduction to Minimax Methods

1. J(u)≥α if u=ρ;

2. There exists v ∈H such that v>ρ and J(v)<α.

Then there exists a Palais–Smale sequence for J at a level c ≥α. If J satisﬁes (PS)

then there exists a critical point at level c.

Proof We deﬁne the minimax class

 ={γ([0, 1]) |γ :[0, 1]→H is continuous,γ(0) =0,γ(1) =v}

and the corresponding minimax level

c = inf

γ([0,1])∈

sup

u∈γ([0,1])

J(u)= inf

γ ∈

max

t∈[0,1]

J (γ (t)).

The sets in  are thus the continuous paths that connect the origin with the point v.

In the last equality we identify γ and γ([0, 1]), with some abuse of notation, as is

customary to do.

If we prove that  is admissible, we apply the minimax principle, and the proof

is done.

It is obvious that c<+∞, since we maximize a continuous functional on a

compact set. We have to prove that c>−∞. To see this, we notice that every γ ∈

satisﬁes γ(0)=0 and γ(1)=v >ρ. Since γ is continuous, there exists

∈[0, 1] such that γ(t

)=ρ. This implies, by assumption 1,

max

t∈[0,1]

J(γ(t))≥J(γ(t

)) ≥α.

As this holds for every γ ∈ , we can take the inﬁmum with respect to γ ∈  to

obtain

c ≥α>0.

We now have to prove that for every ε>0 small enough,  is invariant for all

deformations that ﬁx J

c−2ε

For this, we take any ε>0 so small that

J(0)<c−2ε and J(v)<c−2ε,

which is possible because max{J(0), J (v)}=max{0,J(v)}<α≤c.

Let now η be a deformation that ﬁxes J

c−2ε

and take γ ∈ . We have to prove

that, for all t ∈[0, 1],

η(t, γ ([0, 1])) ∈,

which means that η(t, γ ([0, 1])) =˜γ([0, 1]) for some ˜γ ∈. To this aim we deﬁne

˜γ :[0, 1]→H as ˜γ(s)=η(t, γ (s)).

Of course ˜γ is continuous. Since 0 and v belong to J

c−2ε

, and η ﬁxes this set, we

have

˜γ(0) =η(t,γ (0)) =η(t,0) = 0

4.3 Two Classical Theorems 157

and

˜γ(1) =η(t,γ (1)) =η(t,v) =v.

Hence ˜γ([0, 1]) ∈ , and  is an admissible class. We then apply Theorem 4.2.4

and we conclude the proof. 

Some comments and remarks are in order.

Remark 4.3.2 The term “mountain pass” is justiﬁed by the geometrical properties of

the graph of J . Indeed, it is customary to think of the points 0 and v as two villages.

The assumptions of Theorem 4.3.1 imply that the two villages are separated by a

mountain range: to go from 0 to v one must climb at least at a height α, which is

strictly larger than the heights J(0) and J(v). The minimax class used in the proof

is constructed exactly on this observation: one tries every possible road γ between

0 and v, measures the maximal height reached by the road (max

t∈[0,1]

J(γ(t))), and

tries to minimize the maximal height. At the maximal height of the “optimal” road

is located a (mountain) pass.

Remark 4.3.3 In the ﬁrst part of this chapter, the attention was concentrated on

the change of topology of different sublevels. In the Mountain Pass Theorem, this

change is very simple to detect, and it is exactly what the assumptions describe: there

is a sublevel which is not path-connected. Indeed, take a continuous path γ between

0 and v; this will lie in a sublevel J

, where, for example b = max

t∈[0,1]

J(γ(t)).

This means that 0 and v are in the same connected component of J

. On the con-

trary, 0 and v are not in the same (arcwise) connected component of J

, for every

a<c. Thus J

cannot be deformed into J

Example 4.3.4 A functional that satisﬁes the assumptions of Theorem 4.3.1 is said

to have a “mountain pass geometry”. The functional associated to Problem (4.1),

which we used as a motivation for this discussion, has a mountain pass geometry.

Indeed, J(0) = 0, and one can take for example ρ =C

−

p−2

, α =(

−

p−2

and v any function for which v>ρand J(v)is negative; there are plenty of these

functions since J(λu)→−∞as λ →+∞, for every u ≡0.

A large number of functionals J associated with superlinear problems have a

mountain pass geometry. This is due to the fact that (i) 0 is a strict local mini-

mum and J behaves well enough to prove that 1. in Theorem 4.3.1 is satisﬁed and

(ii) lim

λ→+∞

J(λu)=−∞, which is guaranteed by superlinearity, implies 2.

Remark 4.3.5 As a ﬁnal consideration, we wish to highlight the use of deformations

that ﬁx certain sets in the proof of Theorem 4.3.1. In view of the minimax principle,

the proof of Theorem 4.3.1 reduces to the construction of an admissible minimax

class. Now the class  is deﬁned by three requirements: every γ is continuous and

such that γ(0) = 0 and γ(1) = v. Of course every deformation, being continuous,

preserves the continuity of γ , so the real condition to satisfy is the fact that the

endpoints of γ should be held ﬁxed during the deformations. We have obtained

158 4 Introduction to Minimax Methods

this by the use of deformations that ﬁx a certain sublevel J

c−2ε

where the points

0 and v lie. Of course we do not want that these deformations ﬁx also the sublevel

, otherwise the main argument of the minimax principle breaks down. Summing

up, the reason why the class  is admissible comes from a separation property:

the interesting level c is strictly larger than the levels that have to be kept ﬁxed

during the deformation, which are the levels J(0) = 0 and J(v)< α. This aspect

is present in almost every construction of concrete minimax classes. We will see

another example of this kind in the Saddle Point Theorem below.

The Mountain Pass Theorem has been used an enormous amount of times in the

literature to produce Palais–Smale sequence. Of course its use depends on the fact

that the functional under study possesses the right geometry.

For functionals that do not have the mountain pass geometry, other minimax

classes can be tried. One of the most popular results, that has also been generalized

in many directions, is the Saddle Point Theorem. Roughly speaking, it works for

functionals where the nonlinearity is not superlinear, but “asymptotically” linear at

inﬁnity. We postpone the examples to the next sections.

The proof of the Saddle Point Theorem requires a classical ﬁxed point result, that

we do not prove (for a proof see [2, 17], or [18]).

Theorem 4.3.6 (Brouwer) Let B

={x ∈R

||x|≤R} and let ψ :B

→B

a continuous function. Then ψ has a ﬁxed point.

We are going to use the following equivalent form of the Brouwer Theorem.

Theorem 4.3.7 Let B

={x ∈R

||x|≤R} and let ϕ :B

→R

be a continuous

function such that ϕ(x) =x if |x|=R (ϕ is the identity on ∂B

). Then there exists

x ∈B

such that ϕ(x) =0.

Proof We deﬁne a map ψ :B

→B

ψ(x) =



x −ϕ(x) if |x −ϕ(x)|≤R,

|x−ϕ(x)|

(x −ϕ(x)) if |x −ϕ(x)|>R.

(4.10)

Clearly, the map ψ is well-deﬁned, continuous and ψ(B

) ⊆B

. By the Brouwer

Theorem there exists x ∈B

such that ψ(x)=x.If|x −ϕ(x)|>R, then

x =

|x −ϕ(x)|

(x −ϕ(x)),

and hence |x|=R. This implies ϕ(x) =x, so that |x −ϕ(x)|=0, a contradiction.

So it must be |x − ϕ(x)|≤R, and x = ψ(x) then implies x = x −ϕ(x), namely

ϕ(x) =0. 

Remark 4.3.8 A linear n-dimensional subspace of a Hilbert space is isomorphic

to R

. Therefore the preceding results holds in ﬁnite dimensional subspaces of

Hilbert spaces. This is the form that we are going to use.

4.3 Two Classical Theorems 159

Theorem 4.3.9 (Saddle Point Theorem [42]) Let H be a Hilbert space and assume

that J ∈ C

1,1

(H ). Let E

⊂ H be a n-dimensional linear subspace of H and call

V ⊂ H its orthogonal complement in H , so that H = E

⊕ V . For R>0 let B

be the closed ball in E

with center at the origin and radius R, and let ∂B

be its

boundary in E

. Assume that for some R>0,

max

u∈∂B

J(u)< inf

u∈V

J(u). (4.11)

Then there exists a Palais–Smale sequence for J at a level c ≥inf

J . If J satisﬁes

(PS)

, then there exists a critical point at level c.

Proof We deﬁne the minimax class

 =



ϕ(B

) |ϕ :B

→H is continuous,ϕ(u)=u ∀u ∈∂B



and the corresponding minimax level

c = inf

ϕ∈

max

u∈B

J (ϕ(u)).

The sets in  are continuous n-dimensional surfaces of boundary ∂B

⊂E

As in Theorem 4.3.1, the proof amounts to showing that  is admissible. It is

clear that c<+∞; the interesting part is to check that c>−∞.

Let P :H →E

be the orthogonal projection on E

, so that, u ∈ V if and only

if P(u)=0 and P(u)=u if and only if u ∈E

. Take a generic ϕ ∈ and deﬁne

ψ :B

→E

as ψ(u) =P (ϕ(u)).

The map ψ is of course continuous, and since ϕ is the identity on ∂B

for every u ∈∂B

,ψ(u)=P(ϕ(u))=P(u)=u.

So ψ is a continuous map from B

to E

that is the identity on ∂B

. By Theo-

rem 4.3.7, there exists u

∈ B

such that ψ(u

) = 0, that is P(ϕ(u

)) = 0, which

means ϕ(u

) ∈V . Therefore,

max

u∈B

J(ϕ(u))≥J(ϕ(u

)) ≥ inf

u∈V

J(u)> max

u∈∂B

J(u)>−∞.

As this holds for every ϕ ∈, taking the inﬁmum over  yields

c ≥ inf

u∈V

J(u)> max

u∈∂B

J(u)>−∞.

Nowwehavetoprovethat satisﬁes the invariance properties of Deﬁnition 4.2.3.

Let ε>0 be any number so small that

inf

J −2ε>max

∂B

and take a deformation η that ﬁxes J

c−2ε

We want to prove that if A = ϕ(B

) ∈ , then η(t,A) ∈  for all t ∈[0, 1].To

this aim we deﬁne

ϕ :B

→H as ϕ(u) =η(t, ϕ(u)).

160 4 Introduction to Minimax Methods

Of course ϕ is continuous. We have to check that ϕ(u) =u for all u ∈∂B

. For this,

take u ∈∂B

. By deﬁnition, ϕ(u) =u, and since

J(u)≤max

∂B

J<inf

J −2ε ≤c −2ε,

η(t, u) =u for all t . Therefore

ϕ(u) =η(t, ϕ(u)) =η(t, u) =u,

which shows that

ϕ is the identity on ∂B

. The class  is admissible and the appli-

cation of Theorem 4.2.4 concludes the proof. 

Remark 4.3.10 The preceding proof is identical in its structure to the proof of the

Mountain Pass Theorem. In particular, also in the Saddle Point Theorem the key

property is the separation of the minimax level from the levels that have to be kept

ﬁxed by the deformations. This is the scope of assumption (4.11): the minimax level

c is greater than or equal to inf

J , which is strictly greater than α = max

∂B

J .

Thus we can construct a deformation that ﬁxes J

, where ∂B

lies, and this is

ultimately what makes  admissible.

4.4 Some Applications

We now illustrate the use of the previous theorems by some applications. We have

chosen the simplest cases, but the reader should keep in mind that a very large

number of problems coming from physics, mechanics or geometry can be dealt with

the two results of the preceding section.

The structure of the following results is suggested by that of the Mountain Pass

Theorem and of the Saddle Point Theorem: one checks that the Palais–Smale condi-

tion holds, and, separately, that the “geometry” of the functionals involved ﬁts into

the abstract requirements of the mentioned theorems.

4.4.1 Superlinear Problems

We begin with some superlinear nonlinearities, for which the Mountain Pass Theo-

rem is tailored.

Consider the following problem: given a bounded open set  ⊂R

, N ≥3, and

a nonnegative L

∞

function q deﬁned on , ﬁnd u such that



−u +q(x)u =f(u) in ,

u =0on∂.

(4.12)

A problem of this form has already been discussed in Sect. 2.5. We will now weaken

the assumptions, maintaining the validity of the result.

We set F(t)=



f(s)ds and we consider the following set of assumptions.